UiMiVtKSlTY Of
ILLINOIS LIBRARY
AI URBANA-CHAMPAIGN
STACKS
Digitized by the Internet Archive
in 2011 with funding from
University of Illinois Urbana-Champaign
http://www.archive.org/details/transferfunction496hopw
Faculty Working Papers
College of Commerce and Business Administration
University of Illinois at Urbana-Champaign
FACULTY WORKING PAPERS
College of Commerce and Business Administration
University of Illinois at Urbana-Charapaign
July 12, 1973
THE TRANSFER FUNCTION RELATIONSHIP BETWEEN EARNINGS
AND I IARKET- INDUSTRY INDICES: AN EMPIRICAL STUDY
William S. Hopwood, Assistant Professor of Accountancy
#496
Summary;
The study investigated the hypothesis that univariate ARIIIA forecasts
can be improved upon by using a more general transfer function model
which consists of an ARIIIA model with a market or industry index added.
Statistical analysis of the data indicated that firms' forecasts have
a tendency to perform either very well or very poorly under the
transfer function model as compared to the ARIIIA model (using an
absolute value error metric) .
It was demonstrated that it is possible to develop an a priori rule
for the determination of when the transfer function will outperform
the univariate model. In particular it was found that if a transfer
function outperforms an ARIIIA model for the majority of the first
three periods in the forecast horizon, then there is a significant
probability that it will do the same for periods four through ten.
1 1
• Ml
In recent years there has been an increased emphasis on the forecasting
of accounting earnings. In particular there have been a large number of
studies which have utilized the Box-Jenkins method of forecasting via auto-
regressive integrated moving average (AE.IMA) models. Notable, however, is
that these models are univariate by definition and do not provide for the
statistical modeling of events which occur outside of the earnings series.
The purpose of the study is to explore this limitation by employing a more
general approach which incorporates market and industry index data into
the forecast model.
A primary reason for exploring this more general approach is that
"Financial analysts have long recognized that economy-wide and industry-
wide factors affect the financial numbers of individual firms. Index
models enable quantification of the effects of these factors. Such
quantification can be important when assessing financial trends in a
firm and forecasting financial variables" (Foster, 1978, p. 155).
Specifically, the research method involves the use of the single input
transfer function method developed by Box and Jenkins (1970) . This approach
generalizes the ARIMA model by incorporating an additional predictor variable,
in addition to past earnings, in the form of a market or industry price index.
The motivation for the inclusion of these particular indices is best expressed
by quoting Beaver, Clarke and Wright (1978, pp. 1-2): "Capital market equili-
brium can be characterized as a mapping from states into a set of security
prices. Similarly, earnings are signals from an information system which is
a mapping from states into signals. In general, these could be any relationship
between price and earnings depending upon the nature of the two mappings.
If one assumes that prices and earnings reflect a common set of events it is
-2-
not unreasonable to assume that the two might be positively associated. In
fact, the Ball and 3rcwn study and the empirical evidence provided in cross-
sectional valuation studies provide support for such a view."
The paper will consist of four sections. The first will give a brief ■
discussion of the transfer function and the second will present the research
design. In the third section the results will be presented followed by a
summary and conclusions in section four.
1.0 A Generalization of the Traditional Box-Jenkins Approach
A generalizatian of the ARIMA model is the transfer function (TF) which
has not been generally used but has recently been suggested by Foster (1977).
This forecast method, which generalizes the traditional Box- Jenkins approach,
avoids the univariate limitation. In particular, it generalizes the ARIMA
models by allowing for the simultaneous modeling of the time series properties
of more than one series cf interest. The general form of the transfer func-
tion is (1) y = [^(y^-p yt_2f "■)» f2^Xt » Xt-1 ' '"^'
f3(xt(2), xt-_1<2), ...), fa(xtCu). xt-l(n>' •••> + u(t)3' Note that (1)
completely generalizes the ARIHA. models to rer.ove the univariate restriction.
In particular, f_, f_, ..., f produce a generalization by allowing y to be
modeled as a function of x , x , . . ., x^ , The net result is a very broad
family of models which contain the aRIMA models as a proper subset. In summary,
the transfer function, due to its generality, has the ability to utilize more
data than the ARIMA ric-lcls. Specifically, it can simultaneously utilize the time
series and cross-correlational properties of more than one series for the
purpose of forecasting EPS.
-3-
2.0 Research Design
2.1 The Sample
A sample of thirty airlines was selected. (A list of the sample firms
is presented in Appendix 1.) This industry was chosen because of the avail-
ibility of both an industry index and individual firm EPS for a period suffi-
ciently long to perform the statistical analysis.
The basic requirement for a firm to be selected was the availability
of EPS for 60 quarters. This provided 50 quarters recommended for model
estimation and 10 quarters for forecast error computation. Since only 30
firms in the industry met the selection criteria, the sample was not random.
2.2 General Eypothesis
•The General Hypothesis tested is:
H : ARTMA forecasts of earnings are not improved when an industry
or market index is added to the basic ARIMA model.
H. : H is not true.
A o
This general hypothesis will be opera tionalized by defining an error
metric. In addition the null hypothesis of no interaction between firm and
forecast model (ARIMA vs TF) will be examined.
2.3 Construction and Application of the Forecast Model
Step 1 For each sample firm one univariate and a two bivariate TF models
2
were constructed based on 50 quarters of EPS. The bivariate models
3
were of the form
(2) yt - [f.Cy^, yt_2, .... yt_n), f2(x<k), ,«, ..., x£>), u(t)]
(k - 1,2)
-4-
where x corresponds to the Dow Jones Industrial Index and x£
corresponds to the Standard and Poors* Air Transportation Industry
4
Index. Note that (2) is a special case of (1) above vhere there is
one x variable. This restriction is made because at present there
are a number of unresolved problems with using a TF model which
has more than one x.
Step 2 For each firm forecasts were generated from one to ten periods in
the future from three models: (1) ARIMA (2) TF with the Dow index
added and (3) TF with the Air transportation index added.
3.0 Empirical Results
3.1 Choice of an Error Metric and Associated Statistical Procedure for
Testing the Null Hypothesis
Initially, consideration of an absolute percentage error metric was given;
however, due to near zero demoninators and correspondingly large denominators
a large number of explosive forecast error occured. Because of this problem
it was decided to employ a nonparametric analysis that utilizes simple
absolute forecast error.
One type of nonparametric analysis that has been used in the past is the
performance of a series of separate nonparametric tests for each different
time origin and/or period in the future. This procedure is not used here
because such a method results in making a large number of nonindependent tests,
and in addition it is likely that some of the tests will lead to rejection of
the null by alpha error related chance. Therefore the procedure chosen was
the use of a simple chi square statistic.
-5-
3.2 Test of the Null Hypothesis for Main Effects
The method used to test the null was to create a variable 6. , , for each
firm i, index j and forecast k (i = 1,30, j ■ 1,2, k « 1,10). If a TF forecast
was closer in absolute value to the actual earnings number than the univariate
forecst, then 6. . . was assigned a value of 1 (and 0 otherwise). In those
cases where 6. . equals one we shall say that the TF forecast for firm i,
index j and period k dominates the univariate forecast for the same firm and
period.
The result is that the number of times that a TF forecast dominates
30
for an index j and period k is £ <5, . , . This implies that associated
i=l *■«»
with each index j there is the following vector of frequencies:
30 30 30
[ E ^ t i. S fi± i 2 Z 6i 1 10^
where each element of the vector represents the number of times that the TF
for index j dominates the univariate forecast at time k.
Since there are 30 firms the null hypothesis can be stated that there
is an expected frequency of 15(1/2 x 30) in each cell (i.e., each vector
element). The actual and expected cell frequencies are presented in
Table 1 for both indices 1 and 2 (corresponding to the Dow and Air Trans-
portation indices respectively).
-6-
TABLE 1
Actual and Expected Frequencies for the Number of
Times that the TF Forecast for Index j Dominates
the ARIMA forecast for Period k
Period k
Index j
Frequency
1
2
3
4
5
6
7
8
9
10
Chi
Square
Dow
3 - 1
actual
13
17
21
17
16
19
15
19
13
11
5.41
Air Trans.
J = 2
actual
11
13
20
20
17
19
14
20
13
15
9.79
expected
15
15
15
15
15
15
15
14
13.5
13
For both indices the null hypothesis is not rejected at the a = .1
level. This implies that on the average the TF forecasts are not signifi-
cantly different than the ARIMA forecasts.
3.3 Test of the Null Hypothesis for Interaction Effects
A test of interaction between firms and forecast models was made to
investigate the following question: Do ARIMA models tend to dominate for
some firms and TF models dominate for others?
10
If such an interaction does exist we would expect to find E 6 .
k=l i,:,»k
for a given firm i and index j to be close to 0 or 10 and under the
hypothesis of no interaction we would expect a value of 5. Table 2
presents the results of the test.
-7-
TABLE 2
Chi Square Test of
No Interaction Effect
Index
Test Statistic (29 df)
Approximate Significance
DOW
49.76
.025
AIR
Transportation
37.13
.145
Note that the null is rejected (at o = .1) for the Dow index and
almost rejected at the .1 level for the air transportation index. Note
that this implies that the 6. , , (k = 1,10) are not independent since
the 6. . , for a given firm i and index j have a tendency to be the
i* j >*■
same for all k (i.e., either 0 or 10).
3.4 A Proposed Contingency Rule for the Selection of a TF Model
The results of the interaction tests tend to indicate that in some
cases the univariate modeling procedure can be Improved upon by examining
the performance of a given TF model for a given i and j over some arbitrary
but fixed L periods of the forecast horizon. If the TF tends to dominate
the ARIMA forecasts over the L periods we would expect it to tend to domlm-
ate over the remaining 10 - L periods of the forecast horizon.
In order to oprationalize this hypothesis it was decided to select
those TF models that dominated the corresponding univariate models for at
o
least two out of the first three forecst periods. The variable X. , for
*■* J
firms i and index j was created and assigned a value of 1 if the TF model
dominated the univariate model for the majority of the remaining seven
periods (and 0 otherwise). There were 21 firms that met the selection
-8-
eriterion and under the null hypothesis of equality between the TF and ARIMA
methods we would expect 10.5 (1/2 x 21) firms to have a X , equal to 1.
9
Table 3 presents a test of this hypothesis.
TABLE 3
Test of Equality Between TF and ARIMA Models
on an A Priori Selected Subset of Firms
1
Number of firms for which a TF index
dominated in the majority of the first
3 forecast periods
21
2
Number of the above 21 firms for which
the TF dominated on the majority of
forecast periods 4-10
16
3
Expected frequencies under the
null hypothesis
10.5
Chi Square Statistic with 1 df .
2.88
The statistic of 2.88 is significant at the a = .1 level as
expected.
An additional test was made by counting the total number of times
that the TF dominated over periods 4-10 for the 21 firms selected. Under
the null hypothesis of no difference between the TF and ARIMA methods on
the restricted subpopulation, we would expect the univariate model to
dominate a to:al of 73.5 (21 x 7 x 1/2) times. Table 4 presents a test
of this hypothesis.
-9-
TABLE 4
A Second Test of Equality Between TF and ARIMA Models
on an A Priori Selected Subset of Firms
Total number of times which a priori
selected TF dominated the corresponding
ARIMA forecast
97
Expected frequency under the null
hypothesis
73.5
Chi Square (1 df)
7.514
Again, as expected, the null is rejected (with a = ,1) and the data
indicate that one is better off, on the average, to select the TF index
model if it dominates the univariate in the majority of the first 3
forecast periods.
4.0 SUMMARY AND LIMITATIONS
4.1 Summary and Conclusions
The study investigated the hypothesis that univariate ARIMA forecasts
can be improved upon by using a more general transfer function model which
consists of an ARIMA model with a market or industry index added. Statistical
analysis of the data indicated that firms' forecasts have a tendency to
perform either very well or very poor under the transfer function model as
compared to the ARIMA model (using an absolute value error metric).
It was demonstrated that it is possible to develop an a priori rule
for the determination of when the transfer function will outperform the
-10-
unlvariate model. In particular it vas found that if a transfer function
outperforms an ARIMA model for the majority of the first three periods
in the forecast horizon, then there is a significant probability that it
will do the same for periods four through ten.
4.2 Limitations and Suggestions for Future Research
A primary limitation of the study is that it was restricted to one
industry. It is suggested that the study be replicated in other industries
as well as in the market as a whole.
FOOTNOTES
Some examples of the use of ARIMA models are: Albrecht, Lookabill
and McKeown (1977), Brown and Rozeff (1977). Dopuch and Watts (1972),
Foster (1977), Lorek, McDonald and Patz (1976), and Watts and Zeftwich
(1977).
2
EPS was taken from Moody's Handbook and adjusted for changes in capital
structure. In addition for firms 1, 2, 3, 5, 7, 12, 14, 15, 16, 18, 20,
21, 22, 24, 25, and 29, EPS were computed using information from schedule
B-3 of the Civil Aeronautics Board (CAB) form 41 in conjunction with the
CAB quarterly periodical Air Carrier Financial Statistics.
3
The modeling was done using a program first written by David Pack
of the Ohio State University and modified for local use at the University
of Illinois by James McKeown. In condensed form the models occupy 15
pages and thus are not presented in this study; however they will be
furnished upon written request to the author.
4
Both the Dow and industry indices were computed from averaging monthly
data taken from Security Owners Stock Guide (Standard and Poor's corpor-
ation) .
5 (k)
A major problem is that of modeling cases where the X series are
not independent of each other. The author is presently in the process of
developing an algorithm for modeling these type of series.
When the absolute percentage forecast errors were computed it was
found that approximately 10% of the errors were more than 3 standard devia-
tions from the mean. In addition there were a large number of values
that were a large number of values in excess of 25 standard deviations
from the mean.
The expected cell frequencies for periods 8, 9 and 10 have been slightly
adjusted for missing data. A description of data available for modeling
and testing is presented in Appendix 2.
D
A maximum of one TF was selected for each firm. In the event that
the two TF models were tied, the following rule was applied: (1) if one
TF dominated for three periods and the other for two periods, then the one
dominating for three periods was selected, (2) if both TF's dominated for
two periods, the TF that dominated the other TF for the majority of the
first three periods was selected. The result was that for twelve firms
the Dow index was chosen and for nine firms the transportation index was
chosen.
9
Of the sixteen firms in cell number two, nine were associated with
the Dow index and seven were associated with the transportation index.
BIBLIOGRAPHY
Albrecht, Steve W., Larry L. Lookabill, and James McKeown, "The Time-
Series Properties of Annual Earnings," Journal of Accounting Research
(Autumn 1977), pp. 226-244.
Beaver, William H., Roger Clarke, and William F. Wright, "The Magnitude
of Earnings Forecast Errors," Faculty Working Paper No. 449
(Graduate School of Business, Stanford University: April 1978).
Box, George E. P. and Gwilyn M. Jenkins, Time Series Analysis Forecasting
and Control (Holden-Day, Inc., 1970).
Brown, Lawrence D. and Michael S. Rozeff, "Univariate Time Series Models
of Quarterly Earnings per Share: A Proposed Premier Model," Faculty
Working Paper No. 77-27 (College of Business Administration, the
University of Iowa: October 1977).
Dopuch, Nicholas and Ross Watts, "Using Time-Series Models to Assess the
Significance of Accounting Changes," Journal of Accounting Research
(Spring 1972), pp. 180-194.
Foster, George, Financial Statement Analysis (Prentice Hall, 1978).
Foster, George, "Quarterly Accounting Data: Time-Series Properties and
Predictive-Ability Results," Accounting Review (January 1977),
pp. 1-21.
Lookabill, Larry, "Time Series Properties of Accounting Earnings,"
Accounting Review (October 1976), pp. 724-738.
Lorek, Kenneth, Charles McDonald and Dennis Patz, "Management and Box-
Jenkins Forecast of Earnings," Accounting Review (April 1976),
pp. 321-330.
Watts, Ross L. and Richard W. Leftwich, "The Time Series Properties of
Annual Accounting Earnings," Journal of Accounting Research
(Autumn 1977), pp. 253-271.
M/B/95
APPENDIX 1
LIST OF SAMPLE FIRMS
1. Airlift International
2. Alaska Airlines
3. Aloha Airlines
4. American Airlines
5. Aspen Airways
6. Braniff Airways
7. Caribbean Atlantic Airlines
8. Continental Airlines
9. Delta Airlines
10. Eastern Airlines
11. Tiger International Airlines
12. Frontier Airlines
13. Hawaiian Airlines
14. National Airlines
15. New York Airways
16. North Central Airlines
17. North West Airlines
18. Ozark Airlines
19. Pan American Airways
20. Piedmont Airlines
21. Reeve Airlines
22. SFO Airlines
23. Seaboard World Airlines
24. Southern Airways
25. Texas International Airlines
26. Trans World Airlines
27. UAL (United Airlines)
28. Western Airlines
29. Wien Airlines
30. Allegheney Airlines
Each firm will be subsequently referred to by the identifying number
that precedes it.
APPENDIX 2
DESCRIPTION OF AVAILABLE DATA FOR
FORECAST ERROR ANALYSIS
This appendix gives a firm by firm description of the number of
quarters of data available for forecast error analysis. For each firm
the number of periods in the base period, the origin date for forecasting,
and the number of absolute forecast errors is presented.
firm
number of periods
origin date for
number
in base period
forecasting
1
50
2/74
2
50
2/74
3
50
2/74
4
50
2/74
5
. 30
2/74
6
30
3/74
7 .
- 40
2/74
8
50
2/74
9
50
2/74
10
50
2/74
11
50
2/74
12
50
2/74
13
50
2/74
14
50
2/74
15
50
2/74
16
50
2/74
17
50
2/74
18
50
2/74
19
50
2/74
20
50
2/74
21
50
2/74
22
42
2/74
23
50
2/74
24
50
2/74
25
50
2/74
26
50
2/74
27
50
2/74
28
50
2/74
29
30
2/76
30
50
2/74
number of steps ahead
forecast error was computed
10
10
10
10
10
8
9
7
10
10
10
10
10
10
, 10
10
10
10
10
10
10
10
10
10
10
10
10
10
7
10
For example in the case of firm 1, 50 quarters of data were used in
transfer and univariate estimation, and actual and predicted forecasts
were computed over a 10 period forecast horizon with the first forecast
being for the third quarter of 1974.
y
Live Music Archive
Librivox Free Audio
Metropolitan Museum
Cleveland Museum of Art
Internet Arcade
Console Living Room
Books to Borrow
Open Library
TV News
Understanding 9/11